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A Full-Block S-procedure application to delay-dependent
state-feedback control of uncertain time-delay systems
Corentin Briat, Olivier Sename, Jean-François Lafay
To cite this version:
Corentin Briat, Olivier Sename, Jean-François Lafay. A Full-Block S-procedure application to
delay-dependent state-feedback control of uncertain time-delay systems. IFAC WC 2008 - 17th IFAC World
Congress, Jul 2008, Séoul, South Korea. �hal-00337002�
A full-block S-procedure application to
delay-dependent H
∞state-feedback control
of uncertain time-delay systems
C. Briat∗ O. Sename∗ J.F. Lafay∗∗
∗GIPSA-Lab, Departement of Control Systems (former LAG),
Grenoble Universit´es, ENSIEG - BP46, 38402 Saint Martin d’H`eres -Cedex FRANCE,{ Corentin.Briat,Olivier.Sename}@gipsa-lap.inpg.fr
∗∗IRCCyN - Centrale de Nantes, 1 rue de la No¨e - BP 92101, 44321
Nantes Cedex 3 - FRANCE,Jean-Francois.Lafay@irccyn.ec-nantes.fr
Abstract: This paper deals about the robust stabilization of uncertain systems with time-varying state delays in the delay dependent framework. The system is represented using LFR and stability is deduced from Lyapunov-Krasovskii theorem and full-block S-procedure. We derive sufficient conditions to the existence of a robust H∞ state-feedback control law. As this
sufficient condition is expressed in terms of NMI we propose a relaxation based on the cone-complementary algorithm which is known to lead to good results for such problems. We show the efficiency of our method trough an example.
Keywords: Robust controller synthesis; Robust time-delay systems; Convex optimization 1. INTRODUCTION
Since several years, constant state-delayed systems have been heavily studied since they are often responsible of instability and poor performances (see [Gouaisbaut and Peaucelle, 2006b], [Fridman, 2006b],[Niculescu, 2001] and references therein). More recently, time-varying delays, ap-pearing for instance in communication networks, have sug-gested more and more interest (See [Fridman, 2006a],[He et al., 2004],[Suplin et al., 2006], [Wu, 2003],[Gu et al., 2003] and references therein).
Two kinds of stability results exist in time-delay systems (TDS): delay independent and delay dependent. The first one guarantees stability for every delay from 0 to ∞. This result is actually conservative due to the consideration of delays near +∞. However, delay-independent stabilization may be useful when delayed terms matrices are small in front of non-delayed terms matrices (i.e. the effect of the delay is small). The second one guarantees the system stability over a compact set of delay (e.g [0, hM]) and leads
then to less conservative results. Actually, this type of result better fits the reality because the delays are always bounded from a practical point of view.
In the context of uncertain systems, an useful tool is the H∞synthesis which provides powerful robust analysis and
control design tools. Nevertheless, due to the small gain condition, some conservatism is always induced. That is why scalings are used in order to obtain better results while reducing the conservatism. The scalings generalize the notion of small-gain condition while considering how the system and the uncertainties are connected (and not only their apparent norm as it is used in the classical small-gain theorem). This leads to the notion of well-posedness of feedback systems [Iwasaki and Hara, 1998] which unifies in
a nice unique framework stability and robustness analysis. The scalings in well-posedness analysis are often called separators (full-block multipliers) since they separate the graph of the system and the inverse graph of the uncer-tainty and provide then a necessary and sufficient condi-tion for well-posedness (and hence stability) of feedback systems. The great interest of full block multipliers come from the fact that there is no inertia constraint on a whole space but only on particular subspaces (which is not the case for scaled-small gain theorem where the scalings must be positive definite) [Wu, 2000], [Scherer, 2001], [Wu, 2001].
This papers brings a new method to design state-feedback for TDS using full-block multipliers, and includes the following contributions:
• The type of uncertainties here considered is quite large as the formulation allows to include polytopic uncertainties, matrix bounded uncertainties ... • First we provide sufficient conditions to delay
de-pendent asymptotic stability of uncertain time-delay system. We extend the result of [Gouaisbaut and Peaucelle, 2006a] to the case of uncertain time-delay stability written as an interconnection of the sys-tem and the uncertainty using the linear fractional transformation. The stability with H∞ performance
is given using the so-called full-block S-procedure extending to the delay-dependent case the results in [Wu, 2003].
• Second we derive from this stability lemma a stabi-lizability lemma (or robust state-feedback existence lemma) in terms of Linear and Nonlinear Matrix Inequalities (LMI and NMI).
• As the stabilizability lemma is not tractable we pro-pose a relaxation based on the cone-complementary
algorithm [Ghaoui et al., 1997] which is known to pro-vide good convergence properties in practice (despite of its local convergence).
• Finally, we show the efficiency of our approach and compare it to other methods through several exam-ples.
We consider in this paper systems of the form ˙ x(t) = A (∆)x(t) + Ah(∆)xh(t) +Bu(∆)u(t) +B1(∆)w1(t) z1(t) = C1(∆)x(t) +C1h(∆)xh(t) +D1u(∆)u(t) +D11(∆)w1(t) (1) where x ∈ Rn, x h = x(t − h(t)), h(t) ∈ H , u ∈ Rnu,
w1∈ Rnw, z1∈ Rnz are respectively the system state, the
delayed state, the delay, the control input, the exogenous inputs and the controlled outputs. ∆ ∈ ∆ represents bounded multiplicative uncertainties. The setsH and ∆ are detailed further.
The paper is structured as follows, Section 2 presents use-ful lemmas and system description. Section 3 presents two theorems on robust stability/performance for uncertain system with time varying delays. Section 4 proposes a design method of state-feedback controller through LMIs. The notation is quite standard but let us define Im(A⊥) as
the orthogonal complement of Im(A) (defined as ATA ⊥=
0). A is positive definite (negative definite) on a subspace S means that xTx > 0 for all x ∈S (xTAx < 0 for all
x ∈S ).
2. RECALL AND DEFINITIONS This section briefly recall the necessary background.
2.1 Useful lemmas
Lemma 2.1. Full block S-procedure Suppose S is a sub-space of Rn, T ∈ Rl×n is a full row rank matrix, N is a
compact set of matrices of full row rank. Define the family of subspaces for each U ∈ U
SU =S ∩ Ker(UT ) = {x ∈ S : UT x = 0}
Then the following conditions are equivalent: 1. For any U ∈ U,
N < 0 onSU andSU ∩S0= {0}
where S0 is a fixed subspace of S such that
dim(S0) ≥ k and N ≥ 0 onS0
2. There exists a symmetric matrix R such that for all U ∈ U
N + TTRT < 0 onS and R > 0 on Ker(U) In our caseS represents the nominal system, T specifies the interconnection between the nominal system S and the uncertainty set U. ThereforeSUdenotes the uncertain
system. The lemma renders the implicit conditions based on the uncertain system data to an explicit expression through the full block multiplier R (See [Scherer, 2001] for more details).
2.2 System description
Without loss of generality let us consider system (1) rewritten using the linear fractional transformation pro-cedure as in figure 1: - -∆ TDS w0 z0 z1 w1 ? -y u
Fig. 1. Uncertain linear time delay system
"x(t)˙ z0(t) z1(t) # = "A B 0 B1 C0 D00 D01 C1 D10 D11 # "x(t) w0(t) w1(t) # + "A h C0h C1h # xh(t) + "B u D0u D1u # u(t) w0(t) = ∆z0(t) (2)
The delay is assumed to belong to the set
H :=nh ∈C1(R+, [0, hM]) : hM < +∞, | ˙h| ≤ µ < 1
o
The time-varying uncertainties ∆ belong to the following uncertainty set ∆ := ( s M i δiIdi : |δi| ≤ ki< +∞, di∈ N∗ )
This representation not only captures the size of the uncer-tainties but also their structure. This block representation has been widely used in robust control. It is also possible to extend it to full uncertainty blocks.
Assuming that the uncertain system is well posed (ie. (I − ∆D00 nonsingular for all ∆ ∈ ∆) then it is possible
to represent the uncertain system into an LFT form: "A (∆) A h(∆) B1(∆) C0(∆) C0h(∆) D01(∆) C1(∆) C1h(∆) D11(∆) # = "A A h B1 C0 C0h D01 C1 C1h D11 # + "B 0 D00 D10 # (I − ∆D00)−1∆ [C0 C0h D01]
The full-block S-procedure lemma will translate the stabil-ity and performance tests for uncertain systems into their equivalent formulation using a full-block multiplier. Let us introduce the full block multiplier set F associated with the uncertainty set ∆ ∈ n0× n0.
F :=F ∈ S2n0 :∆T I n0 F ∆ In0 > 0, ∀∆ ∈ ∆
Since ∆ is infinite dimensional the previous constraint leads to an infinite number of constraints, which is not implementable. However, this can be relaxed under certain conditions on the uncertainty and the multiplier structure. For more details on these relaxations, the readers should refer to [Scherer, 2001, Wu, 2003, Scherer and Hol, 2006]. 3. DELAY DEPENDENT STABILITY CRITERIUM This section propose a robust delay-dependent lemma for uncertain time-delay system with rational dependence
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
onto the parameters. Generally, only few papers consider (see for instance Wu [2003]) such a type of dependence while generally treat the case of polytopic uncertainties Gouaisbaut and Peaucelle [2006b], Suplin et al. [2006] or norm bounded uncertain matrices Xu et al. [2006]). The robust delay-dependent stability lemma for uncertain time-delay systems with time-varying delays is presented below:
Theorem 3.1. The system (2) is asymptotically stable with a L2 induced norm on channel w1→ z1 lower than γ > 0
for all ∆ ∈ ∆ if there exist symmetric matrices P, Q, R > 0 and a scaling matrix F ∈F such that
Π + ΘTF Θ < 0 (4) where Θ = 0C 0 I 0 0 0 0 C0h D00 D01 0 0 and Π is defined by (3).
Proof : A sketch of a proof is developed in appendix A. Remark 3.1. It is worth noting that the proof embeds a rigorous application of the full-block S-procedure lemma (as done in [Wu, 2003]). This explains the length and the weight of the proof.
Note also that in the present stability lemma, we do not introduce any slack variable as usually done in the literature. The only matrix introduced is the topological separator [Iwasaki and Hara, 1998] added by the full-block S-procedure which is theoretically lossless. This matrix is radically different than ’slack’ variables used in many delay-dependent results (see [Park, 1999, Xu and Lam, 2007]). It plays a fundamental role in the stability of interconnections ([Iwasaki and Hara, 1998, Scherer, 2001]).
4. ROBUST H∞ STATE FEEDBACK DESIGN
In that section we propose a result to design a state feedback with H∞performance achievement for uncertain
state-delayed systems with time-varying delays.
We consider now the closed-loop system obtained from the interconnection of system (2) and the control law u(t) = Kx(t): "x(t)˙ z0(t) z1(t) # = "A cl B0 B1 C0cl D00 D01 C1cl 0 D11 # "x(t) w0(t) w1(t) # + "A h C0h 0 # xh(t) w0(t) = ∆z0(t) (5)
where Acl= A+BuK, C0cl= C0+D0uK and C1cl= C1+
D1uK. Note that we consider here that the controlled
output is certain (i.e. D10= 0) and does not contain any
delayed-state (C1h = 0). This is a weak assumption since
on one hand there is no need to control the delayed-state and on the other hand the controlled output is a virtual output for design purpose and hence does not involve any uncertainties.
The stabilization problem formulation is here expressed using the backward adjoint of a time-delay system [Ben-soussan et al., 2006].
4.1 Adjoint system
The backward adjoint of the closed-loop system (5) is defined as [Bensoussan et al., 2006]
˙˜ x(t) ˜ z0(t) ˜ z0h(t) ˜ z1(t) = ATcl C0clT C1hT C1clT ATh BT0 DT00 0 0 0 0 0 DT00 0 B T 0 BT1 DT01 0 D11T 0 ˜ x(t) ˜ w0(t) ˜ w0h(t) ˜ w1(t) ˜ xh(t) (6) ˜w0(t) ˜ w0h(t) = ¯∆ ˜z˜z0(t) 0h(t) , ¯∆ = ∆ ⊕ ∆ (7) Due to the fact that the matrix C0h is non-zero then we
obtain this particular form for the adjoint system. As the output signal z0(t) is the original system contains
a delayed-signal (i.e. xh(t)), then we obtain an adjoint
system with a delay-input signal. In order to account for it in the stability analysis, we back-propagate delay-operator through the uncertainty. This explains why the adjoint system involves a delayed uncertainty and supplementary signals (i.e. w0h(t)).
It is worth noting that when either the matrix acting on the delayed-state is certain or it is the only uncertain ma-trix, the adjoint system admits a more simple expression where the uncertainty matrix does not need to be repeated. In all cases, the multipliers set becomes
¯ F := F ∈ R4n0 :∆¯T I2n0 F ¯ ∆ I2n0 > 0, ∀ ¯∆ ∈ ¯∆ (8) 4.2 State-Feedback existence lemma
We prove in this section the state-feedback existence lemma.
Lemma 4.1. There exists a robust state-feedback control law of the form u(t) = Kx(t) such that the closed-loop system (5) is asymptotically stable for all ∆ ∈ ∆ if there exist symmetric matrices P, Q, R > 0, a scalar γ > 0 and F ∈ ¯F such that M1+ Θ1F ΘT1 < 0 (9) N2T(M2+ Θ2F ΘT2)N2< 0 (10) where M1 is defined in (11), M2 in (12), Θ1 = 0 0 B0 0 0 0 0 B0 I 0 D00 0 0 I 0 D00 0 0 0 0 0 0 0 0 , Θ2 = 0 0 B0 0 I 0 D00 0 0 0 0 0 0 0 0 B0 0 I 0 D00 0 0 0 0 0 0 0 0 and N2 = KerBT u D T 0u D T 1u ⊕ I.
Proof : See appendix B.
As the matrix inequality (9) is nonlinear due to the term −h−1MP R−1P , this lemma cannot be easily solved. We reformulate it into the following form.
Lemma 4.2. There exists a robust state feedback control of the form u(t) = Kx(t) such that the closed-loop system (5) is asymptotically stable for all ∆ ∈ ∆ if there exist symmetric matrices P, Q, R, S, T > 0, symmetric definite matrices W, Z < 0 and a scalar γ > 0 and F ∈ ¯F such that
Π = ATP + P A + Q − h−1MR ? ? ? ? ? AThP + h−1MR −(1 − µ)Q − h−1 MR ? ? ? ? BT0P 0 0 ? ? ? BT1P 0 0 −γInw ? ? C1 0 D10 D11 −γInz ? hMRA hMRAh hMRB0 hMRB1 0 −hMR (3) M1:= Q − h−1MR − h−1MP R−1P ? ? ? ? ? h−1MR −(1 − µ)Q − h−1MR ? ? ? ? 0 0 0 ? ? ? 0 0 0 0 ? ? 0 0 0 0 −γInz ? B1T 0 DT01 0 DT11 −γInw (11) M2:= AP + P AT+ Q − h−1MR ? ? ? ? ? ? C0P 0 ? ? ? ? ? C1P 0 −γInz ? ? ? ? AhP + h−1MR 0 0 −(1 − µ)Q − h −1 MR ? ? ? C0hP 0 0 0 0 ? ? BT1 DT01 DT11 0 0 −γInw ? hMRAT hMRC0T hMRCT1 hMRATh hMRC0hT 0 −hMR (12) M3:= Q − h−1MR + h−1MZ ? ? ? ? ? h−1MR −(1 − µ)Q − h−1MR ? ? ? ? 0 0 0 ? ? ? 0 0 0 0 ? ? 0 0 0 0 −γInz ? B1T 0 D01T 0 DT11 −γInw (13) M3+ Θ1F ΘT1 < 0 (14) N2T(M2+ Θ2F ΘT2)N2< 0 (15) W T T −S ≤ 0 (16)
with ZW = I, RS = I and P T = I, where M2 is defined
in (12), M3 in (13),
Proof : The proof is identical as in [Chen and Zheng, 2006]. This problem is obviously non-convex due to equalities ZW = I, RS = I and P T = I but as in [Chen and Zheng, 2006] such a problem can be approximatively solved using the cone complementary algorithm (see [Ghaoui et al., 1997]).
Algorithm 1. Cone complementary algorithm (1) Fix γ = γ0.
(2) Fix k = 0, γ and find P0, Q0, R0, S0, T0, W0, T0
satis-fying (14), (15), (16).
(3) Find (Pk+1, Tk+1, Rk+1, Sk+1, Wk+1, Zk+1) that solves
minP,Q,R,S,T ,W,Z Tr(P Tk+PkT +RSk+RkS +W Zk+ WkZ) with (14) (15), (16) and P I I T ≥ 0 R I I S ≥ 0 W I I Z ≤ 0 (17) (4) If it is feasible
• If the optimal value is 6n reduce then update γ to a smaller value and go to step 2
• else k ← k + 1 and go to step 3.
else if γ ≥ γmax then exit else update γ to a larger
value and go to step 2. 4.3 Controller Construction
We provide here methods to construct the controller from the solutions of the stabilizability lemma.
Explicit Construction This method uses a simple algo-rithm borrowed from [Iwasaki and Skelton, 1995].
Algorithm 2. (1) Find λ > 0 such that Φ := (λUTU −
M2− Θ2F ΘT2)−1> 0 where U =BT u D T 0u D T 1u 0 0 0 0
(2) Compute K = −λU ΦVT(V ΦVT)−1 with V =
[P 0 0 0 0 0 hMR]
This method allows to construct explicitly the controller and is parametrized by the term λ hence there exists an infinite number of stabilizing controllers satisfying H∞
closed-loop performances. Note that λ can be easily found using SDP. For a full parametrization see for instance [Iwasaki and Skelton, 1994] and references therein. Implicit Construction This part explains how to con-struct the controller in an implicit manner. In this case, the controller is found as a solution of a SDP and allows to add supplementary constraints.
Lemma 4.3. The controller matrix K is found while solv-ing the followsolv-ing SDP
min K,ν,t J (K, ν, t) M2+ Θ2F ΘT2 + U TKV + VTKTU ≤ −tI L(K, ν, t) < 0 (18)
where J is a cost to minimize, ν, t are additional decision terms and L supplementary convex constraints.
Notice that if the constraints L are too strong then it may be not possible to find a feasible solution to this LMI problem even if the stabilizability problem is feasible. In this case, the constraints should be make weaker.
5. EXAMPLE Consider now the following system
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
˙ x(t) = h−1.3 0.2 0.2 −1 i x(t) + h1 0 i u(t) + h1 1 i w(t) + h−0.6 −0.5 −0.5 −0.6 i + h1 0 1 −0.8 i ρ + h−0.9 1 0.1 −1 i ρ2 xh(t) z(t) = h0 1 0 0 i x(t) + h0 0.1 i u(t) (19)
We aim compare our approach to the approach proposed in [Suplin et al., 2006] which gives very good results for polytopic systems. However this system cannot be directly expressed as a polytope due to the polynomial dependence onto the uncertain parameter ρ ∈ [−1, 1]. One way to consider it as a polytope is to make the following restrictive change of variables: ρ1:= ρ and ρ2:= ρ2∈ [0, 1] (it would
be also possible to construct a ellipsoid containing the family of matrices parametrized by ρ on ). It is obvious that such an approximation is not good since we consider aberrant points (for instance ρ1 = 1 and ρ2 = 0 which
never occurs). It is possible to reduce the size of the polytope but the epigraph of the function f (ρ1) = ρ21 has
to remain convex which is source of conservatism. With no polytope reduction, the approach of [Suplin et al., 2006] does not lead to find a stabilizing controller while using the approach presented in this paper we obtain γ = 8.765 with a state-feedback gain K = [35.0747 −19.3227]. We can easily imagine that a system looses stabilizability for some values into the epigraph of f (·) but not onto the image set of f (·). In that case, the polytopic approach would lead to more conservative result than methods using the linear fractional transformation.
6. CONCLUSION
We have proposed a new method of H∞ robust
stabil-ity/performance analysis for uncertain system with time-varying delays through full block multipliers in the delay-dependent framework in terms of a finite number of LMIs. A robust state-feedback existence lemma is derived from stability condition. As the resulting conditions are non-convex we provide tractable conditions using the cone complementary algorithm which is known to be efficient in practice but does not guarantee global convergence. From the solutions of the stabilizability conditions, we provide how construct the controller matrix either using an explicit formulation or an implicit one through SDP. Due to linear fractional representation of the uncertain system it is possible to tackle a wider class of uncertain system (such as polynomial or even rational dependence onto uncertain parameters) and we show that the proposed approach leads to better results.
Appendix A. PROOF OF THEOREM 3.1 Consider the following Lyapunov-Krasovskii functional
V = xT(t)P x(t) + Z t t−h(t) xT(θ)Qx(θ)dθ . . . + Z 0 −hM Z t t+θ ˙ x(η)TR ˙x(η)dηdθ
and the uncertain system
˙
x(t) =A (∆)x(t) + Ah(∆)x(t − h(t)) +B1(∆)w(t)
z(t) =C (∆)x(t) + Ch(∆)x(t − h(t)) +D11(∆)w(t)
Denoting xh(t) := x(t−h(t)) and computing the derivative
V along the trajectories solutions of the system (we drop the dependence on time and ∆ for ease of simplicity) we obtain ˙ V ≤ (A x + Ahxh+B1w)P x + (?)T + xTQx − (1 − µ)xThQxh+ hMx˙TR ˙x − Z t t−h(t) ˙ x(θ)TR ˙x(θ)dθ | {z } I
Using the Jensen’s inequality on the integral term leads to I ≤ −h−1M Z t t−h(t) ˙ x(θ) !T R Z t t−h(t) ˙ x(θ) !
Note that the first equation is not defined for h(t) = 0 but it can easily be shown that the bound on I is well defined in ti with h(ti) = 0. The same inequality for constant
time-delays is also presented in [Gouaisbaut and Peaucelle, 2006b].
Replacing the term ˙x by its explicit expression leads to the quadratic form (where we drop the dependency on ∆ for ease of simplicity XTΞX < 0 (A.1) where X =xT xT h wT T and Ξ11=ATP + PA + Q + hMATRA − h−1MR Ξ21=AhTP + hMAhTRA + h −1 MR Ξ22= −(1 − µ)Q + hMAhTRAh− h−1MR Ξ31=B1TP +B1TRA Ξ32= hMBT1RAh Ξ33= hMBT1RB1
This LMI is infinite dimensional due to the dependence on the uncertainty function ∆. This quadratic form can be expressed as (?)TΘ I 0 0 A (∆) Ah(∆) B1(∆) 0 I 0 I 0 0 −I I 0 A (∆) Ah(∆) B1(∆) < 0 where Θ = 0 P P 0 ⊕−(1 − µ)Q 0 0 Q ⊕−h −1 MR 0 0 hMR . To specify the input/output H∞ constraint, we add
to the Lyapunov-Krasovskii functional derivative the in-put/output constraint s(w, z) = −γwTw + γ−1zTz where γ > 0 is a positive scalar. Then we obtain
(?)TΘγ I 0 0 A (∆) Ah(∆) B1(∆) 0 I 0 I 0 0 −I I 0 A (∆) Ah(∆) B1(∆) 0 0 I C1(∆) C1h(∆) D11(∆) < 0
where Θγ = 0 P P 0 ⊕−(1 − µ)Q 00 Q ⊕−h −1 MR 0 0 hMR ⊕ −γIw 0 0 γ−1Iz .
Then we apply the full-block S-procedure, expand the expression of the obtained LMI and finally perform Schur complement onto quadratic term
− C1T hMATR 0 hMAThR DT10 hMB0TR DT11 hMB1TR −γ−1I 0 0 −h−1MR−1 (?)T
and we obtain LMI (4). This proof is then complete. Appendix B. PROOF OF LEMMA 4.1
First inject matrices of augmented adjoint system (6) into LMI of statement 3 of theorem 3.1. Then note that the inequality (4) can be rewritten as
M2+ Θ2F ΘT2 + U TKV + VTKTU < 0 where M2 is defined in (12), Θ2= 0 0 B0 0 I 0 D00 0 0 I 0 D00 0 0 D10 0 0 0 0 B0 0 0 0 0 0 0 0 0 , U = BT u D T 0u D T 1u 0 0 0 0, V = [P 0 0 0 0 0 hMR].
Then using projection lemma [Apkarian and Adams, 1998] this nonlinear inequality is equivalent to two underlying inequalities:
N1T(M2+ Θ2F ΘT2)N1< 0 (B.1)
NT
2 (M2+ Θ2F ΘT2)N2< 0 (B.2)
LMI (B.2) is obviously (10) with N2 = Ker(U ). Now
consider P P1+ hMRP2= 0 then we have
N1:= Ker(V ) = "P 1 0 0 I P2 0 #
As P, R > 0 (hence nonsingular), there exists an infinite number of couples (P1, P2) such that P P1+ hMRP2 = 0.
Let P1 = I and we obtain P2 = −h−1MR
−1P . Use this
basis to project in inequality (B.1). The result is matrix inequality (9) modulo some rows/columns permutations. This concludes the proof.
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17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
